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    On Euler characteristic and fundamental groups of compact manifolds

    Bing-Long Chen and Xiaokui Yang

    Abstract. Let M be a compact Riemannian manifold, π : M̃ → M be the universal

    covering and ω be a smooth 2-form on M with π∗ω cohomologous to zero. Sup-

    pose the fundamental group π1(M) satisfies certain radial quadratic (resp. linear)

    isoperimetric inequality, we show that there exists a smooth 1-form η on M̃ of lin-

    ear (resp. bounded) growth such that π∗ω = dη. As applications, we prove that

    on a compact Kähler manifold (M,ω) with π∗ω cohomologous to zero, if π1(M)

    is CAT(0) or automatic (resp. hyperbolic), then M is Kähler non-elliptic (resp.

    Kähler hyperbolic) and the Euler characteristic (−1) dimR M

    2 χ(M) ≥ 0 (resp. > 0).

    Contents

    1. Introduction 1

    2. Hyperbolic fundamental groups 6

    3. General fundamental groups 12

    4. Quadratic radial isoperimetric inequality 16

    References 20

    1. Introduction

    In differential geometry, there is a well-known conjecture due to H. Hopf (e.g.

    [Yau82, Problem 10]):

    Conjecture 1.1 (Hopf). Let M be a compact, oriented and even dimensional Rie-

    mannian manifold of negative sectional curvature K < 0. Then the signed Euler

    characteristic (−1)n2 χ(M) > 0, where n is the real dimension of M .

    For n = 4, Conjecture 1.1 was proven by S. S. Chern ([Che55]) (who attributed it to J.

    W. Milnor). Not much has been known in higher dimensions. This conjecture can not

    be established just by use of the Gauss-Bonnet-Chern formula (see [Ger76, Kle76]).

    I. M. Singer suggested that in view of the L2-index theorem an appropriate vanishing

    theorem for L2-harmonic forms on the universal covering of M would imply the con-

    jecture (e.g. [Dod79]). In the work [Gro91], Gromov introduced the notion of Kähler

    hyperbolicity for Kähler manifolds which means the Kähler form on the universal

    cover is the exterior differential of some bounded 1-form. He established that the

    1

    http://arxiv.org/abs/1711.03309v2

  • B.-L. Chen and X.-K. Yang On Euler characteristic and fundamental groups of compact manifolds

    L2-cohomology groups of the universal covering of a Kähler hyperbolic manifold are

    not vanishing only in the middle dimension. Combining this result and the covering

    index theorem of Atiyah, Gromov showed (−1)n2 χ(M) > 0 for a Kähler hyperbolic manifold M . One can also show that a compact Kähler manifold homotopic to a

    compact Riemannian manifold of negative sectional curvature is Kähler hyperbolic

    and the canonical bundle is ample([CY17]).

    When the sectional curvature of the manifold is non-positive, it is natural to con-

    sider whether (−1)n2 χ(M) ≥ 0 holds. It should be noted that when the sectional curvature of a compact Kähler manifold is nonpositive, [JZ00] and [CX01] proved

    independently that the vanishing theorem of Gromov type still holds and the Euler

    characteristic satisfies (−1)n2 χ(M) ≥ 0. Actually, they proved that the results also hold if the pulled-back Kähler form on the universal covering is the exterior differential

    of some 1-form with linear growth, and such manifolds are called Kähler non-elliptic

    ([JZ00]) and Kähler parabolic ([CX01]) respectively. In the sequel, for simplicity, we

    shall use one of these notions, e.g., Kähler non-elliptic. Moreover, Jost and Zuo pro-

    posed an interesting question in [JZ00, p. 4] that whether there are some topological

    conditions to ensure the manifolds to be Kähler non-elliptic. One can also propose

    the following

    Question 1.2. Let M be a compact Riemannian manifold, π : M̂ → M a Galois covering and G the group of covering transformations. Let ω be a closed q-form

    (q ≥ 2) on M such that [π∗ω] = 0 in HqdR(M̂). Find a (q − 1)-form η on M̂ of least growth order (in terms of the distance function on M̂) such that π∗ω = dη.

    It is clear that the growth order of η does not depend on the choices of the metrics on

    M , and it should depend on the geometry of the covering transformation group G.

    Recall that by a theorem of Gromov, a discrete group G is hyperbolic if and only if it

    satisfies a linear isoperimetric inequality. An expected answer for Question 1.2 would

    be a relation between certain isoperimetric inequality of the covering transformation

    group G and the least growth order of η.

    We need some well-known notions of discrete groups to formulate an isoperimetric

    inequality in our setting. Suppose G = 〈S|R〉 is a finitely presented group, where S is a finite symmetric set generating G, S = S−1, and R is a finite set (relator set) in the free group FS generated by S. The word metric on G with respect to S is defined

    as

    (1.1) dS(a, b) = min{n : b−1a = s1s2 · · · sn, si ∈ S}. For a word w = s1s2 · · · sn, its length L(w) is defined to be n. If the word w = s1 · · · sn ∈ FS representing the identity e in G, there are reduced words v1, · · · , vk on S such that

    (1.2) w = k∏

    i=1

    viriv −1 i , ri or r

    −1 i ∈ R

    2

  • On Euler characteristic and fundamental groups of compact manifolds B.-L. Chen and X.-K. Yang

    as elements in FS . The combinatorial area Area(w) of w is the smallest possible k for

    equation (1.2).

    Definition 1.3. We say a finitely presented group G = 〈S|R〉 satisfies a radial isoperi- metric inequality of degree p, if there is a constant C > 0 such that for any word

    w = s1 · · · sn ∈ FS of length L(w) = n representing the identity e in G, we have

    (1.3) Area(w) ≤ C n∑

    i=1

    (dS(w(i), e) + 1) p−1 ,

    where w(i) = s1 · · · si is the i-th subword of w and w(i) ∈ G is the natural image (from FS to G) of the word w(i).

    It is easy to see that the above definition is independent of the choice of the generating

    set S. For p = 1, this definition is the same as the usual linear isoperimetric inequality

    Area(w) ≤ CL(w).

    For p > 1, Definition 1.3 is stronger than the usual isoperimetric inequality. Actually,

    the radial isoperimetric inequality (1.3) can imply

    (1.4) Area(w) ≤ C(diam(w) + 1)p−1L(w) ≤ CL(w)p.

    We obtain in Theorem 3.1 a complete answer to Question 1.2 for q = 2. For simplicity,

    we only formulate the polynomial growth case which has many important applications.

    Theorem 1.4. Let M be a compact Riemannian manifold, π : M̂ → M a Galois covering with covering transformation group G and H1dR(M̂) = 0. Let ω be a closed

    2-form on M such that [π∗ω] = 0 ∈ H2dR(M̂ ). Assume that the group G satisfies the radial isoperimetric inequality (1.3) of degree p ≥ 1. Then there exists a smooth 1-form η on M̂ such that π∗ω = dη and

    (1.5) |η|(x) ≤ C(d M̂ (x, x0) + 1)

    p−1

    for all x ∈ M̂ where C is a positive constant and x0 ∈ M̂ is a fixed point.

    Let’s explain briefly the key ingredients in the proof of Theorem 1.4 and demonstrate

    the significance of the radial isoperimetric inequality (1.3). Let G = 〈S|R〉 be the finitely presented covering transformation group. The condition [π∗ω] = 0 implies π∗ω = dη for some smooth 1-form η on M̃ .

    (i) We show that there exists a constant C such that for any closed curve α on M̃ ,

    we can construct a word w ∈ FS representing the identity and approximating α such that

    (1.6)

    ∣∣∣∣ ∫

    α

    η

    ∣∣∣∣ ≤ C(L(α) + Area(w)).

    3

  • B.-L. Chen and X.-K. Yang On Euler characteristic and fundamental groups of compact manifolds

    (ii) By using the radial isoperimetric inequality (1.3), we prove

    (1.7)

    α

    η ≤ C ∫

    α

    (d M̂ (x, x0) + 1)

    p−1ds

    for any closed curve α.

    (iii) η can be regarded as a “bounded linear functional” L on the space of closed

    curves with a suitably defined norm. Then we use the Hahn-Banach theo-

    rem and Whitney’s local flat norm ([Whi57]) to find another bounded linear

    functional L̃ whose restriction on the space of closed curves is L. Moreover,

    L̃ is represented by a differential form η̃ with measurable coefficients with

    π∗ω = dη̃ (in the current sense) and

    (1.8) |η̃|(x) ≤ C(d M̂ (x, x0) + 1)

    p−1, a.e.

    (iv) We use the heat equation method to deform η̃ to a smooth one with the

    desired bound in (1.5).

    The radial isoperimetric inequality (1.3) is the key ingredient in step (ii), and it could

    not work for usual isoperimetric inequality for degree p > 1. With the inequality

    (1.7) in hand, step (iii) is classical (e.g. [Gro81, Gro98, Sik01, Sik05]). The smooth-

    ing process in step (iv) is natural in the view point of PDE since ω = dη is preserved

    under the specified heat equation (3.16) and the estimate (1.5) follows from standard

    apriori estimate of heat equations.

    As an application of Theorem 1.4, we obtain

    Theorem 1.5. Let (M,ω) be a compact Kähler manifold and π : M̃ → M be the universal covering. Suppose [π∗ω] = 0 ∈ H2dR(M̃ ). If π1(M) satisfies the radial